Just building on what others have said. You will need a waste carriers license if you plan on transporting the waste, you don't need it just to store waste. It is free if you only transport waste produced by yourself, but has a cost for transporting waste produced by others. If you are buying or selling the waste, you will also need a dealers registration, which is part of your waste carriers license.
The waste carriers license system is currently in the process of being reformed, so you might have to track the changes:
https://www.gov.uk/government/consultations/waste-carrier-broker-and-dealer-system-reform/outcome/government-response-and-summary-of-responses#contents
You and the waste producer will have to determine and agree if the waste is hazardous or non-hazardous, and assign it a European Waste Catalogue (EWC) Code. You can check the available EWC codes to get some guidance. There are a few websites that display these; I personally use https://dsposal.uk/browse/ewc. If that doesn't help, the Environment Agency can be contacted to help classify waste, but there will be a charge.
Please ensure you have waste transfer notes for waste transactions. These are a legal requirement and can be issued per transfer or as a season ticket for up to one year. Guidance below:
https://www.gov.uk/dispose-business-commercial-waste/waste-transfer-notes
I will add additional information on environmental permitting below as it wasn't clear to me if you have one.
To take in waste from other producers, you will need either an environmental permit or a waste exemption. A permit is a large undertaking and I wouldn't recommend it if you don't plan on processing the waste on your sites.
An exemption is much more reasonable if you only plan on storing waste. There are some exemptions that allow for the processing of waste, but they have some limitations. The Environment Agency and the Government have guidance that can be found with a google search.
Please note, waste exemptions are also being reformed. Some of the changes can be found on the below link:
https://thecompliancepeople.co.uk/updates/resources/changes-to-waste-exemptions/
I'm happy to provide any additional information that I can. I only work in waste management and I'm not a consultant.
Regards,
Mathew